Asymptotics for configural location estimators
Let \(x_ 1,...,x_ n\) be independent real-valued observations from the distribution function (d.f.) F(x-\(\mu)\); it is assumed that F(\(\cdot)\) is symmetric around 0, \(0<F(x)<1\) for all \(x\in {\mathbb{R}}\), and has a probability density function f(\(\cdot)\) with respect to Lebesgue measure. When sampling from a symmetric situation as above, the Pitman estimator of \(\mu\) is given by \[ T_ F(x_ 1,...,x_ n)=\int^{\infty}_{- \infty}r\prod^{n}_{i=1}f(x_ i+r)dr/\int^{\infty}_{- \infty}\prod^{n}_{i=1}f(x_ i+r)dr. \] This estimator has the smallest mean-square-error among all location equivariant estimators, where T(\(\cdot)\) is location equivariant, if \(T(x_ 1+r,...,x_ n+r)=T(x_ 1,...,x_ n)+r\), \(r\in {\mathbb{R}}.\) In this paper, the main interest is centered at the behavior of the so- called bioptimal or compromise location estimators when the sample is coming from a d.f. G(\(\cdot)\) symmetric around 0. With d.f.'s \(F_ 1(\cdot)\) and \(F_ 2(\cdot)\) satisfying the same assumptions as the d.f. F(\(\cdot)\) above, a compromise estimator \(T_{F_ 1,F_ 2,\alpha}(x_ 1,...,x_ n)\) is defined by \[ T_{F_ 1,F_ 2,\alpha}(x_ 1,...,x_ n)= \] \[ \alpha w_{F_ 1}(x_ 1,...,x_ n)T_{F_ 1}(x_ 1,...,x_ n)+(1-\alpha)w_{F_ 2}(x_ 1,...,x_ n)T_ F(x_ 1,...,x_ n), \] where \(0<\alpha <1\); for \(k=1,2\), \(T_{F_ k}(x_ 1,...,x_ n)\) are given as \(T_ F(x_ 1,...,x_ n)\) above, and the weights \(w_{F_ k}(x_ 1,...,x_ n)\) are given by \[ w_{F_ k}(x_ 1,...,x_ n)= \] \[ \int^{\infty}_{- \infty}\prod^{n}_{i=1}f_ k(x_ i+r)dr/\int^{\infty}_{- \infty}\{\alpha \prod^{n}_{i=1}f_ 1(x_ i+r)+(1- \alpha)\prod^{n}_{i=1}f_ 2(x_ i+r)\}dr, \] k\(=1,2\). In many applications, \(F_ 1(\cdot)\) is chosen to be of the Gaussian shape, and, on the basis of various considerations, \(F_ 2(\cdot)\) is chosen as a heavy tailed distribution; for example, a distribution with Pareto tails. Under certain overly nice conditions, which do not include, for example, distributions with Pareto tails, the asymptotic behavior of the weights is established; namely, it is shown that \(w_{F_ 2}(x_ 1,...,x_ n)/w_{F_ 1}(x_ 1,...,x_ n)\) converges to 0 a.s. The same result is also proved under weaker conditions. It follows that the compromise estimator is equivalent to the Pitman estimator. Furthermore, expressions for the asymptotic variances of \(T_{F_ 1}(x_ 1,...,x_ n)\) and \(T_{F_ 2}(x_ 1,...,x_ n)\) are given. Finally, if the d.f. G(\(\cdot)\) is chosen to be \(G(x)=(1-\epsilon)\Phi (x)+\epsilon H(x)\), where \(\Phi\) (\(\cdot)\) is the d.f. of the standard normal distribution, \(H(x)+H(-x)=1\) for all x's and H(\(\cdot)\) puts all its mass outside the interval [-k,k], but is otherwise arbitrary, and \(0\leq \epsilon \leq 0.5\), it is shown that the asymptotic variance of the compromise estimator is \(\leq\) the asymptotic variance of Huber's minimax estimator.
scientific article; zbMATH DE number 3974060